NDLI: Existence, characterization and stability of Pansu spheres in sub-Riemannian 3-space forms

Content Provider	Springer Nature Link
Author	Rosales, César Hurtado, Ana
Copyright Year	2015
Abstract	Let M be a complete Sasakian sub-Riemannian 3-manifold of constant Webster scalar curvature $$\kappa $$ . For any point $$p\in M$$ and any number $$\lambda \in {\mathbb {R}}$$ with $$\lambda ^2+\kappa >0$$ , we show existence of a $$C^2$$ spherical surface $$\mathcal {S}_\lambda (p)$$ immersed in M with constant mean curvature $$\lambda $$ . Our construction recovers in particular the description of Pansu spheres in the first Heisenberg group (Pansu, Conference on differential geometry on homogeneous spaces (Turin, 1983), pp 159–174, 1984) and the sub-Riemannian 3-sphere (Hurtado and Rosales, Math Ann 340(3):675–708, 2008). Then, we study variational properties of $$\mathcal {S}_\lambda (p)$$ related to the area functional. First, we obtain uniqueness results for the spheres $$\mathcal {S}_\lambda (p)$$ as critical points of the area under a volume constraint, thus providing sub-Riemannian counterparts to the theorems of Hopf and Alexandrov for CMC surfaces in Riemannian 3-space forms. Second, we derive a second variation formula for admissible deformations possibly moving the singular set, and prove that $$\mathcal {S}_\lambda (p)$$ is a second order minimum of the area for those preserving volume. We finally give some applications of our results to the isoperimetric problem in sub-Riemannian 3-space forms.
Ending Page	3227
Page Count	45
Starting Page	3183
File Format	PDF
ISSN	09442669
e-ISSN	14320835
Journal	Calculus of Variations and Partial Differential Equations
Issue Number	3
Volume Number	54
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2015-07-12
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Sub-Riemannian geometry Immersions (minimal, prescribed curvature, tight, etc.) Systems Theory, Control Calculus of Variations and Optimal Control; Optimization Analysis Theoretical, Mathematical and Computational Physics
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis

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